mathsGrade 9-102 min read

Straight-line graphs: everything from y = mx + c

One equation describes every straight line. Learn to read the gradient and intercept off it — and to find the equation of a line from two points — and the whole topic opens up.

Every straight line in the plane can be written as

y = mx + c,

and the entire topic is reading meaning into those two letters: $m$ and $c$ . Get what each one does and you can sketch any line, or find its equation, in seconds.

What m and c tell you

$c$ is the $y$ -intercept — where the line crosses the $y$ -axis. It's the value of $y$ when $x = 0$ , so it's literally where the line starts.
$m$ is the gradient — how steep the line is. It's the "rise over run": how far up you go for each step to the right.

m = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1}

A positive $m$ slopes uphill; a negative $m$ slopes down; a bigger number is steeper.

Finding the equation from two points

Given two points, say $(1, 3)$ and $(3, 9)$ :

Gradient first: $m = \dfrac{9 - 3}{3 - 1} = \dfrac{6}{2} = 3$ .
Then the intercept: sub one point into $y = 3x + c$ . Using $(1,3)$ : $3 = 3(1) + c \Rightarrow c = 0$ .
Write it out: $y = 3x$ .

Parallel and perpendicular

Relationship	Gradient rule
Parallel lines	Same gradient $m$
Perpendicular lines	Gradients multiply to $-1$ (negative reciprocals)

So a line perpendicular to $y = 2x + 1$ has gradient $-\tfrac{1}{2}$ — flip it and change the sign.

Gradient before intercept, always. Find $m$ from the two points first, then substitute a point to get $c$ . Trying to eyeball $c$ from a sketch is where the marks leak away.

Last revised 15 May 2025.